SOLUTION: Find all zeros of the following polynomial. Write the polynomial in factored form. f(x) = 4x^3 + 7x^2 + 6x - 2

Algebra ->  Polynomials-and-rational-expressions -> SOLUTION: Find all zeros of the following polynomial. Write the polynomial in factored form. f(x) = 4x^3 + 7x^2 + 6x - 2      Log On


   

Question 765440: Find all zeros of the following polynomial. Write the polynomial in factored form.
f(x) = 4x^3 + 7x^2 + 6x - 2
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

f%28x%29+=+4x%5E3+%2B+7x%5E2+%2B+6x+-+2..writ 7x%5E2 as 8x%5E2-x%5E2 and 6x as 8x-2x
f%28x%29+=+4x%5E3%2B8x%5E2%2B8x-x%5E2-2x-2.....group
f%28x%29+=+%284x%5E3-x%5E2%29%2B%288x%5E2-2x%29%2B%288x-2%29
f%28x%29+=+x%5E2%284x-1%29%2B2x%284x-1%29%2B2%284x-1%29
f%28x%29+=+%284x-1%29%28x%5E2%2B2x%2B2%29
solutions:
%284x-1%29%28x%5E2%2B2x%2B2%29=0
if 4x-1=0 => 4x=1 => x=1%2F4
now use quadratic formula to find remaining two roots from %28x%5E2%2B2x%2B2%29=0

x+=+%28-b+%2B-+sqrt%28+b%5E2-4%2Aa%2Ac+%29%29%2F%282%2Aa%29+
x+=+%28-2+%2B-+sqrt%28+2%5E2-4%2A1%2A2+%29%29%2F%282%2A1%29+
x+=+%28-2+%2B-+sqrt%28+4-8+%29%29%2F2+
x+=+%28-2+%2B-+sqrt%28+-4+%29%29%2F2+
x+=+%28-2+%2B-+2i%29%2F2+
x+=+-2%2F2+%2B-+2i%2F2+
x+=+-1+%2B-+i+
roots are:
x+=+-1+%2B+i+
and
x+=+-1+-+i+

so, you have one real and two complex roots

+graph%28+600%2C600%2C+-10%2C10%2C+-10%2C+10%2C+4x%5E3+%2B+7x%5E2+%2B+6x+-+2%29+